Three-dimensional trigonometry problems can be very
hard and complex, mainly because it’s sometimes hard to visualise what
the question is asking. If there is a diagram given in the question it can
make things easier, but it can still be challenging thinking about exactly what
you need to do to find an answer. It pays to keep a calm head and think about
exactly what you need to find and all the things that you know.

Usually these 3D questions will require more than one
step to solve. For instance, take this classic pyramid type question, where
you need to find the length AE:

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3D diagrams are usually more complex than 2D ones and
have more information present in the diagram. This one shows a pyramid. The
four dashes through the four sides of the base show that all four sides are
equal in length, and the right angle shown in one corner of the base tells us
for certain that the base is a square (if all four sides were equal length but
there were no angles marked it could be a parallelogram base).

The vertical height of the pyramid (not the slant
height) is 10 units. Also we know that the angle formed between the slant line
up the middle of a side and the base is 72°.

Here’s where we ask ourselves – what are we trying to
find? In this case, it’s given straight to us in the question – the length of
side AE. How can we find out this length? Well, there are a couple of
triangles which it forms part of, shown here:

Both these triangles are right angled triangles,
meaning if we could find the length of both other sides, we could find the
length of side AE using Pythagoras’ Theorem. The second triangle option looks
attractive, because one of the sides is the vertical part of the triangle,
which we already know is 10 units long. So let’s work with this triangle. We
just need to find the length of its short side now.

The short side of the triangle forms half of the
diagonal of the square base. At this stage it pays to add some more
letters into our diagram so we can more easily describe what we’re doing in any
calculations. We can also draw a square representing the base, and a triangle
representing the one we’re working with in the diagram:

So to recap – we know that EF is 10 units long. If we
can work out the length of AF, then we can use Pythagoras’ Theorem to work out
the length of AE. So how do we find out AF? Well, look at the square base diagram.
If we can work out the side length of the square base, we can use Pythagoras’
Theorem to work out the length of AF. So let’s work on the side length of the
base.

The only unused piece of information in the question is
the 72° angle. Let’s have a look at that triangle:

Its short side forms part of the square base. In fact,
its short side is exactly half the length of the base’s sides. We can
use trigonometry to work out the length of FG, which is half the side length of
the base. Here’s the triangle we’re working with:

If we go back to our diagram of the base, we now know
the length of FG:

The two lengths marked with arrows are the same length
as FG, so we can use Pythagoras’ Theorem to work out the length AF:

We can go back to our original triangle with the side
AE in it that we’re trying to find. We now know the lengths of the other two
sides (AF and EF), so it’s just Pythagoras’ Theorem to find AE:

Notice how many 2-dimensional diagrams I drew
along the way. It makes it easier when you start off with the 3-dimensional
diagram, and then work at bits of it, by drawing 2-dimensional parts of it and
working on each part individually. By labelling the corners and important
parts of the 3-dimensional diagram, you can also label the points in these
simpler 2-dimensional diagrams. Having labels in both diagrams will help you keep
track of where the simple 2D diagrams fit into the 3D one.

Find the height of the large tree. Bob knows that the
angle between the closest point to him in a row of trees and the large tree
is 37 degrees. He also knows that the angle of elevation from the ground to
the top of the tall tree is 13 degrees.

Solution

This is a pretty common type of question – there’s an
animal or something in the top of a tree like in the diagram:

Sometimes the hardest bit of these types of questions
is working out which angles are in which plane. For instance, in the
diagram, the 13 degree angle is an angle of elevation between the
bottom and top of the large tree. The 37 degree angle is a bit more
ambiguous. What hopefully the question has is a statement something like
this:

Bob knows that the angle between the closest point to
him in a row of trees and the large tree is 37 degrees.

This clarifies that the 37 degree angle is in the ground
plane. The right angle at the end of the 70 m line also shows that that
point in the trees is the closest point that the question is talking about.

Usually the question is asking you to find a length
or angle somewhere in the diagram. In this case, the question is asking you
to find the height of the large tree. Now we’ve got a right angled
triangle with one side that is the height of the tree, and we also know
another angle – 13°. All we need is one more side length in that triangle,
and we can work out our answer. Let’s label our 3D diagram before we go any
further:

The triangle we’re talking about is triangle ACD:

Side CD is the height of the large tree. All we need
is the length of another side, either AD or AC. Let’s work on side AC, since
it’s in the ground plane, unlike AD.

Now side AC is also part of another triangle, the
triangle in the ground plane with the 37° angle and 70 metre side:

For this triangle we have a known side length and we
have a known angle. This means we can find out any other side length or
angle in it. What we want is the length of side AC. We can use cosine
to find this out:

Now here’s a trick. Rather than working out the
value of AC, let’s just remember this expression for what AC’s length
is. Now, back to our other triangle, armed with this new information:

So we now know AC, and we’re trying to find CD. We
need to use tangent:

This is the part where we substitute in our
remembered expression for what the length of AC is: